Variant of the Maximum-Modulus Principle

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Homework Statement


Show that the maximum principle holds with [tex]\phi :\mathbb{C}\rightarrow\mathbb{R}[/tex], 

[tex]\phi(z) = (\textrm{Re}(z))^4 + (\textrm{Im}(z))^4[/tex],

in place of the modulus:

If [tex]U \subset \mathbb{C}[/tex] is open and connected, [tex]f : U \rightarrow \mathbb{C}[/tex] is holomorphic and [tex]p \in U[/tex] is such that

[tex]\phi(f(p)) \geq \phi(f(z)) [/tex]

then [tex]f[/tex] is constant.

Can one also replace the modulus by the sine of the modulus, i.e. the function with [tex]\phi(z) = sin(|z|)[/tex]?


Homework Equations


If [tex]z=x+iy[/tex]

[tex]\phi(z)=x^4+y^4[/tex]


The Attempt at a Solution


I tried using the open mapping theorem and the fact that

[tex]f(U)\subseteq\{z|\phi(z)\leq\phi(f(p))\}=\{z|x^4+y^4\leq M\}[/tex]

where [tex]M=\phi(f(p))[/tex], but was not successful.
 

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